Detection Thresholds in Very Sparse Matrix Completion

We study the matrix completion problem: an underlying m× n matrix P is low rank, with incoherent singular vectors, and a random m× n matrix A is equal to P on a (uniformly) random subset of entries of size dn. All other entries of A are equal to zero. The goal is to retrieve information on P from the observation of A. Let A be the random matrix where each entry of A is multiplied by an independent { 0 , 1 } -Bernoulli random variable with parameter 1/2. This paper is about when, how and why the non-Hermitian eigen-spectra of the matrices A1(A-A1)∗ and (A-A1)∗A1 captures more of the relevant information about the principal component structure of A than the eigen-spectra of AA and AA. We show that the eigenvalues of the asymmetric matrices A1(A-A1)∗ and (A-A1)∗A1 with modulus greater than a detection threshold are asymptotically equal to the eigenvalues of PP and PP and that the associated eigenvectors are aligned as well. The central surprise is that by intentionally inducing asymmetry and additional randomness via the A matrix, we can extract more information than if we had worked with the singular value decomposition (SVD) of A. The associated detection threshold is asymptotically exact and is non-universal since it explicitly depends on the element-wise distribution of the underlying matrix P. We show that reliable, statistically optimal but not perfect matrix recovery, via a universal data-driven algorithm, is possible above this detection threshold using the information extracted from the asymmetric eigen-decompositions. Averaging the left and right eigenvectors provably improves estimation accuracy but not the detection threshold. Our results encompass the very sparse regime where d is of order 1 where matrix completion via the SVD of A fails or produces unreliable recovery. We define another variant of this asymmetric principal component analysis procedure that bypasses the randomization step and has a detection threshold that is smaller by a constant factor but with a computational cost that is larger by a polynomial factor of the number of observed entries. Both detection thresholds allow to go beyond the barrier due to the well-known information theoretical limit d≍ log n for exact matrix completion found in the literature.